It is OK. I invited my parents last year and they got the visa. I also changed my status before: F2->F1->H-1B. Don't worry! :)

Yes you can.I transferred from F-2 to F-1, now H-1b and my parents had no problem visiting us!

there're many successful cases, however, visa for parent is very unpredictable now. My friend, US citizen, can't even get her dad to be here for her graduation commencement. Her dad used to visit the US very often.

Just be prepared.